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   "source": [
    "# VARMAX models\n",
    "\n",
    "This is a brief introduction notebook to VARMAX models in statsmodels. The VARMAX model is generically specified as:\n",
    "$$\n",
    "y_t = \\nu + A_1 y_{t-1} + \\dots + A_p y_{t-p} + B x_t + \\epsilon_t +\n",
    "M_1 \\epsilon_{t-1} + \\dots M_q \\epsilon_{t-q}\n",
    "$$\n",
    "\n",
    "where $y_t$ is a $\\text{k_endog} \\times 1$ vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import statsmodels.api as sm\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "dta = sm.datasets.webuse('lutkepohl2', 'https://www.stata-press.com/data/r12/')\n",
    "dta.index = dta.qtr\n",
    "endog = dta.loc['1960-04-01':'1978-10-01', ['dln_inv', 'dln_inc', 'dln_consump']]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model specification\n",
    "\n",
    "The `VARMAX` class in statsmodels allows estimation of VAR, VMA, and VARMA models (through the `order` argument), optionally with a constant term (via the `trend` argument). Exogenous regressors may also be included (as usual in statsmodels, by the `exog` argument), and in this way a time trend may be added. Finally, the class allows measurement error (via the `measurement_error` argument) and allows specifying either a diagonal or unstructured innovation covariance matrix (via the `error_cov_type` argument)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 1: VAR\n",
    "\n",
    "Below is a simple VARX(2) model in two endogenous variables and an exogenous series, but no constant term. Notice that we needed to allow for more iterations than the default (which is `maxiter=50`) in order for the likelihood estimation to converge. This is not unusual in VAR models which have to estimate a large number of parameters, often on a relatively small number of time series: this model, for example, estimates 27 parameters off of 75 observations of 3 variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "exog = endog['dln_consump']\n",
    "mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(2,0), trend='n', exog=exog)\n",
    "res = mod.fit(maxiter=1000, disp=False)\n",
    "print(res.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the estimated VAR model, we can plot the impulse response functions of the endogenous variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ax = res.impulse_responses(10, orthogonalized=True).plot(figsize=(13,3))\n",
    "ax.set(xlabel='t', title='Responses to a shock to `dln_inv`');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 2: VMA\n",
    "\n",
    "A vector moving average model can also be formulated. Below we show a VMA(2) on the same data, but where the innovations to the process are uncorrelated. In this example we leave out the exogenous regressor but now include the constant term."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(0,2), error_cov_type='diagonal')\n",
    "res = mod.fit(maxiter=1000, disp=False)\n",
    "print(res.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Caution: VARMA(p,q) specifications\n",
    "\n",
    "Although the model allows estimating VARMA(p,q) specifications, these models are not identified without additional restrictions on the representation matrices, which are not built-in. For this reason, it is recommended that the user proceed with error (and indeed a warning is issued when these models are specified). Nonetheless, they may in some circumstances provide useful information."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(1,1))\n",
    "res = mod.fit(maxiter=1000, disp=False)\n",
    "print(res.summary())"
   ]
  }
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